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ERCIM News No.50, July 2002


Introduction to the Special Theme: ERCIMathematics

by Tom Koornwinder

Compared to special themes in earlier issues of ERCIM News the present theme, mathematics, is unusually wide and unusually old: the profession of mathematician is believed to be older than all but two other professions. So what ambition should we have to cover this theme in a few dozens of pages? Why not just refer to the marvellous 1200 pages book ‘Mathematics Unlimited - 2001 and Beyond’, B. Engquist & W. Schmidt (eds.), Springer? Well, the theme is not just mathematics, but ERCIMathematics. This may be a restriction which is not very drastic. In the scientific fields of competence of the ERCIM institutes are listed. Six out of sixteen areas listed in the rough subdivision of fields belong to mathematics:

  • algebra, analysis and geometry
  • combinatorics
  • number theory
  • control and system theory
  • stochastics
  • numerical mathematics and differential equations,

while four other areas (theoretical computer science, fluid dynamics, electromagnetism, and operations research) heavily use mathematics. In reality, as most readers will know, ERCIMathematics is not so comprehensive as suggested by the above list. There is a heavy bias to applied mathematics and to mathematics in interdisciplinary research, most prominently in connection with ICT.

Furthermore, there has been a dynamics in ERCIM institutes of putting less emphasis and focus on mathematics. Even members with a traditionally strong mathematics component, like INRIA or CWI, have experienced that the expansion in ICT was not matched by a proportional boost in mathematics research. Nevertheless there is still a considerable volume of mathematics research going on in ERCIM institutes. Part of it is not so explicitly visible because it is embedded in projects under a more general name. To be sure, several developments in ICT pose problems that require a mathematical approach, and for which no ready mathematical solution is at hand. In fact, these mathematical problems can be very challenging and may, when eventually solved, also open up new fruitful directions in pure mathematics.

In preparing the Call for this issue, the mathematical landscape in ERCIM has been taken into account. In particular the following possible topics for papers were mentioned:

  • Computer algebra
  • Computational geometry (eg, in connection with robotics)
  • Automatic theorem proving (including software correctness proofs)
  • Computer-assisted mathematics
  • Interactive books (eg, Abramowitz & Stegun)
  • Mathematics on the Web
  • e-Math (electronic education, eg, online courses)
  • Dynamical systems (non-linear equations)
  • Image understanding
  • Signal processing
  • Complexity theory
  • Optimisation
  • Queuing theory
  • Computational mathematics (numerical methods).

This is reflected in the following selection of articles, mainly coming from ERCIM institutes. The selection is neither exhaustive nor representative in the sense that the institutes with the strongest mathematics component also should have the highest weight. It just represents the spontaneous reaction on the call for contributions to this special issue. It turned out that four fields were represented with several articles: Internet/Web, Numerical Mathematics, Image Processing, and Symbolic Computation. Some other fields were represented with one article.

Some features in this harvest can be observed. First, not surprisingly, almost all papers have something to do with computers: math describing computers or computer networks, math developed by heavy use of computers, math aimed at delivering software. Second, many papers are about projects of larger size and time horizon. Third, many papers describe computer packages in an earlier or later state of completion.

Of course, with the topics mentioned and with the necessity that papers are short and do not become too technical, one cannot expect the traditional Definition-Lemma-Theorem style of mathematical papers. Also, one will not find in this volume proofs taken from the Book of Heaven. What one can pick up by browsing through and reading in this journal is a flavour of important recent ideas and concepts in applied and computer oriented mathematics, and also some trends and some surprising common concepts and keywords in different papers. Sometimes the math is mainly figuring as a useful and productive language, elsewhere deep math is used or developed for new applications. Jonathan Borwein, guest contributor from Canada, describes how experimental mathematics can lead to the discovery of mathematical gems.
The World Wide Web has had an enormous impact on society and on professional and daily life. It is also a fascinating source of mathematical questions how the Web develops as an organism and how the Web can be managed. Several papers here deal with these issues.

Stochastics has permeated the whole society and much of science. Also in this issue the importance of stochastic ideas, techniques and models becomes clear.

One editorial problem, which did not play a role so much with many earlier themes in ERCIM News, is the special way in which mathematics is expressed. The usual way of expression in ERCIM News is ordinary language combined with pictures. Mathematicians additionally want to (and have to) use formulas for making clear their ideas. In fact, this Special Theme Editor loves formulas, believes that they can have great beauty, and that they are much more efficient and precise than words for making clear what is meant. To paraphrase an old proverb, popular in image processing circles: one formula is worth a thousand words (the length of this Introduction, which I did not try to condense into one formula). Still the permanent rules of ERCIM News cannot be easily put away, and they were introduced because of the wide readership of ERCIM News, including researchers far off from mathematics, and managers. This trade-off between mathematical precision, as expressed in formalisation, and readability for a public not particularly trained in reading formulas, has led to the compromise for this Special Theme that formulas are presented as much as possible in separate boxes.

In preparing this special issue, invaluable help was provided by Henk Nieland from CWI, ERCIM Local Editor for The Netherlands.

Please contact:
Tom H. Koornwinder,
Korteweg-de Vries Institute for Mathematics,
University of Amsterdam
(temporarily also CWI), The Netherlands.